#494743

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2072 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,16,12,17,8,14)(7,18,20,33,22,34,26,36)(9,24,19,28,10,15,23,32)(13,27,30,25,31,35,29,21);(5,7,19,35,12,22,23,27)(6,13,28,36,17,31,32,33)(8,20,10,25,11,26,9,21)(14,29,15,18,16,30,24,34);(2,4)(5,8,12,11)(6,14,17,16)(7,20,22,26)(9,19,10,23)(13,29,31,30)(15,32,24,28)(18,33,34,36)(21,35,25,27);(5,9,12,10)(6,15,17,24)(7,21,22,25)(8,23,11,19)(13,18,31,34)(14,28,16,32)(20,27,26,35)(29,36,30,33)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284111, #72285800, #72285823
maximal normal subgroup of	#72284111, #72285800, #72285823

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401939);